Optimal. Leaf size=74 \[ \frac {2 (a+b x)^{5/2} \sqrt [5]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (\frac {1}{5},\frac {5}{2};\frac {7}{2};-\frac {d (a+b x)}{b c-a d}\right )}{5 b \sqrt [5]{c+d x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {70, 69} \[ \frac {2 (a+b x)^{5/2} \sqrt [5]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (\frac {1}{5},\frac {5}{2};\frac {7}{2};-\frac {d (a+b x)}{b c-a d}\right )}{5 b \sqrt [5]{c+d x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 69
Rule 70
Rubi steps
\begin {align*} \int \frac {(a+b x)^{3/2}}{\sqrt [5]{c+d x}} \, dx &=\frac {\sqrt [5]{\frac {b (c+d x)}{b c-a d}} \int \frac {(a+b x)^{3/2}}{\sqrt [5]{\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}} \, dx}{\sqrt [5]{c+d x}}\\ &=\frac {2 (a+b x)^{5/2} \sqrt [5]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (\frac {1}{5},\frac {5}{2};\frac {7}{2};-\frac {d (a+b x)}{b c-a d}\right )}{5 b \sqrt [5]{c+d x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 73, normalized size = 0.99 \[ \frac {2 (a+b x)^{5/2} \sqrt [5]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (\frac {1}{5},\frac {5}{2};\frac {7}{2};\frac {d (a+b x)}{a d-b c}\right )}{5 b \sqrt [5]{c+d x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 4.08, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )}^{\frac {3}{2}}}{{\left (d x + c\right )}^{\frac {1}{5}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{\frac {3}{2}}}{{\left (d x + c\right )}^{\frac {1}{5}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.19, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right )^{\frac {3}{2}}}{\left (d x +c \right )^{\frac {1}{5}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{\frac {3}{2}}}{{\left (d x + c\right )}^{\frac {1}{5}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,x\right )}^{3/2}}{{\left (c+d\,x\right )}^{1/5}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b x\right )^{\frac {3}{2}}}{\sqrt [5]{c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________